Examine some contracted 1--s functions. Let
where n is the number of primitive functions in the contracted function
, and 
are the contraction coefficients. The product
can be written
Since the bracketed term contains a product of two polynomials, only two types of terms can result; the square of each uncontracted function and the products of all different pairs of uncontracted functions. Take an example where n is three:
In this case, as in all others, there are only two types of terms of which the integral needs to be taken. They may be written and evaluated as
It is realized that 1. above can be obtained by setting i=j in 2., and henceforth only the general case need be considered. Generalizing to arbitrary n is straightforward, and so the normalization of contracted gaussian functions can proceed as
thus the normalization constant for the entire contraction will be
General contractions (of arbitrary angular momentum) are a tad worse, but if we assume all of the contracted functions to be of the same angular momentum
The product in brackets in equation 1.40 we've encountered before. Analogous to equation 1.7, the general form for one integral in the double sum is
Thus the product can be written as one sum if we're clever. The self overlap is then
Calling l+m+n = L, the angular momentum of the shell, and solving for N,
